You did not say anything about looking for vacuum leaks.
You really need to do this first. It could be that simple. Vacuum leaks are usually easy to find and fix.
Could there be a problem with the air pump?
Yes the air pump has a clutch on it and only operates for a short while at startup. The clutch lining wears out and you then have metal on metal grinding sound. No you cannot replace only the clutch. Yes the air pump is expensive. What some people do is just disconnect the electric plug that controls it. That solves the grinding noise. But there is also a vacuum valve on it. The hose nipple on mine rusted and created a vacuum leak. I replaced mine and then the clutch went out.
No change in the idle when I spray carb cleaner around the injectors.
This will only expose injector seal or intake leaks that will make your car run lean not rich.
I don't know. Do a search and I am sure you will find an answer.
I noticed you are new here. Welcome aboard. There is a lot of smart people here that can answer almost anything. The first thing to do is use the search engine. It works pretty good. There are a lot of questions all ready answered.
How many miles on the car?
Are you using platinum plugs?
If over 120000 has the timing chain been replaced?
Hope this helps